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# The square root of integer x is equal to the sum of the cubes of y and

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Math Expert
Joined: 02 Sep 2009
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The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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19 Jul 2017, 23:49
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28% (01:52) correct 72% (01:46) wrong based on 134 sessions

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The square root of integer x is equal to the sum of the cubes of y and z. If the squares of y and z are each less than 10, what is the greatest possible value of x?

A. 2916
B. 3981
C. 3982
D. 3999
E. 4000

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Joined: 02 Aug 2009
Posts: 6253
The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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20 Jul 2017, 03:03
1
Bunuel wrote:
The square root of integer x is equal to the sum of the cubes of y and z. If the squares of y and z are each less than 10, what is the greatest possible value of x?

A. 2916
B. 3981
C. 3982
D. 3999
E. 4000

Hi...
Let the values be √10 each. here the values will be just LESS than 10, so our ACTUAL answer will be slightly less than the answer we get here
So sum of cubes is $$2*10^{\frac{3}{2}}$$
Square of this is $$4*10^3=4000$$..

But the actual ans will be slightly LESS than 4000 and an integer..
So 3999
D
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The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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25 Jul 2017, 10:46
chetan2u wrote:
Bunuel wrote:
The square root of integer x is equal to the sum of the cubes of y and z. If the squares of y and z are each less than 10, what is the greatest possible value of x?

A. 2916
B. 3981
C. 3982
D. 3999
E. 4000

Hi...
Let the values be √10 each. here the values will be just LESS than 10, so our ACTUAL answer will be slightly less than the answer we get here
So sum of cubes is $$2*10^{\frac{3}{2}}$$
Square of this is $$4*10^3=4000$$..

But the actual ans will be slightly LESS than 4000 and an integer..
So 3999
D

Chetan - Thanks for responding. Although, I wish you hadn't used "yellow" to highlight the text against an almost yellow background. Nevertheless, here is my approach and I don't know if I arrived at the wrong answer.

Given,

$$\sqrt{x}$$ = $$y^3$$ + $$z^3$$

And is also given,

$$y^2$$ < 10
$$z^2$$ < 10

So, the biggest value that y and z could each take is 3. Since, the question asks for the highest value, I have assumed the highest value for y and z (i.e, 3).

$$\sqrt{x}$$ = $$3^3$$ + $$3^3$$
=> $$\sqrt{x}$$ = 27 + 27
=> $$\sqrt{x}$$ = 54 and squaring both sides, we get:
x = $${54}^2$$
x = 2916.

Please correct if I am wrong.
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Re: The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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25 Jul 2017, 11:28
1
1
Bunuel wrote:
The square root of integer x is equal to the sum of the cubes of y and z. If the squares of y and z are each less than 10, what is the greatest possible value of x?

A. 2916
B. 3981
C. 3982
D. 3999
E. 4000

We can create the following equation:

√x = y^3 + z^3

Since the squares of y and z are each less than 10, y < √10 and z < √10, which implies y^3 < 10√10 and z^3 < 10√10.

Thus:

√x < 10√10 +10√10

√x < 20√10

x < 4000

Since x is an integer, the greatest value of x is 3999.

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Re: The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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25 Jul 2017, 12:27
JeffTargetTestPrep wrote:
.
.
.
y < √10 and z < √10, which implies y^3 < 10√10 and z^3 < 10√10
.
.
.

Beats me why I should "root" the y instead of assuming a possible highest value for y in $$y^2$$ e.g., y= 3 ?
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Re: The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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25 Jul 2017, 12:34
1
Blackbox wrote:
JeffTargetTestPrep wrote:
.
.
.
y < √10 and z < √10, which implies y^3 < 10√10 and z^3 < 10√10
.
.
.

Beats me why I should "root" the y instead of assuming a possible highest value for y in $$y^2$$ e.g., y= 3 ?

Y and z need not be integers

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Math Expert
Joined: 02 Aug 2009
Posts: 6253
Re: The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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25 Jul 2017, 18:55
Blackbox wrote:
chetan2u wrote:
Bunuel wrote:
The square root of integer x is equal to the sum of the cubes of y and z. If the squares of y and z are each less than 10, what is the greatest possible value of x?

A. 2916
B. 3981
C. 3982
D. 3999
E. 4000

Hi...
Let the values be √10 each. here the values will be just LESS than 10, so our ACTUAL answer will be slightly less than the answer we get here
So sum of cubes is $$2*10^{\frac{3}{2}}$$
Square of this is $$4*10^3=4000$$..

But the actual ans will be slightly LESS than 4000 and an integer..
So 3999
D

Chetan - Thanks for responding. Although, I wish you hadn't used "yellow" to highlight the text against an almost yellow background. Nevertheless, here is my approach and I don't know if I arrived at the wrong answer.

Given,

$$\sqrt{x}$$ = $$y^3$$ + $$z^3$$

And is also given,

$$y^2$$ < 10
$$z^2$$ < 10

So, the biggest value that y and z could each take is 3. Since, the question asks for the highest value, I have assumed the highest value for y and z (i.e, 3).

$$\sqrt{x}$$ = $$3^3$$ + $$3^3$$
=> $$\sqrt{x}$$ = 27 + 27
=> $$\sqrt{x}$$ = 54 and squaring both sides, we get:
x = $${54}^2$$
x = 2916.

Please correct if I am wrong.

Hi,

Sorry for the colour. Didn't realise it.

You are WRONG when you read it as an integer.
It is nowhere given as an integer.
So if √10=3.16227766.., the value could be 3.16227765..
Hence we find answer by taking it as 10 and finding one value lesser INTEGER.

Hope it helps
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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25 Jul 2017, 19:52
perfect squares less than 10 are 1 ,4,9 for which value of y,z could be 1 ,2 ,3

but for greatest possible value of x we will take y= z= 3

now √x = y^3+z^3 =3^3 + 3^3

√x = 27+27 = 54

x= 54^2
x=2916

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Re: The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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26 Jul 2017, 06:56
Perfect squares less than 10 are 1 ,4,9 for which value of y,z could be 1 ,2 ,3
but for greatest possible value of x we will take y= z= 3
now √x = y^3+z^3 =3^3 + 3^3
√x = 27+27 = 54
x= 54^2
x=2916
Option A.

Problem states square root of integer x is equal to the sum of the cubes of y and z, hence to achieve that we cannot have y & z to be non integer values as we frame the equation x=y^3+z^3=10√10+10√10, as √10 when simplified would give a decimal component as well. Or in other words y & z cannot be √10 each.
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Re: The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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26 Jul 2017, 07:53
Hi FB2017

Although I did the same as you did, but I was wrong. below is explanation-

we are not saying x=10√10+10√10 rather it is x<10√10+10√10 i.e x<20√10 (Note: if we take x=20√10 then we will get x=4000)

as x<20√10 we are saying that x<4000 i.e x=3999 as x should be an integer.

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The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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14 Aug 2017, 02:41
2
Bunuel wrote:
The square root of integer x is equal to the sum of the cubes of y and z. If the squares of y and z are each less than 10, what is the greatest possible value of x?

A. 2916
B. 3981
C. 3982
D. 3999
E. 4000

$$\sqrt{x} = y^3 + z^3$$
$$y^2 <10 , z^2 <10$$
$$-> y <\sqrt{10}, z <\sqrt{10}$$

So, $$\sqrt{x} < \sqrt{10}^3 + \sqrt{10}^3$$
$$\sqrt{x} < 20*\sqrt{10}$$
$$x < (20*\sqrt{10})^2$$
$$x<4000$$
$$So x = 3999$$

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The square root of integer x is equal to the sum of the cubes of y and  [#permalink]

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14 Aug 2017, 03:23
1
Ans is clearly D

consider max values
y^2=10 means y^6 =1000 and same for z^6=1000. or y=z
x^0.5 = y^3+z^3
square both sides
x= y^6+z^6+2(yz)^3
= 1000+1000+2(z^2)^3
= 1000=1000+2(z^6)
=1000+1000+2(1000)
= 4000
but y and z are less than 10
therefore x is less than 4000 and integer value less than 4000 is 3999.

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The square root of integer x is equal to the sum of the cubes of y and &nbs [#permalink] 14 Aug 2017, 03:23
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